The square root of 2 (approximately 1.4142) is a positive

^{''n''} th terms of a

Let $S\; =$ shorter length and $L\; =$ longer length of the sides of a sheet of paper, with

:$R\; =\; \backslash frac\; =\; \backslash sqrt$ as required by ISO 216. Let $R\text{'}\; =\; \backslash frac$ be the analogous ratio of the halved sheet, then

:$R\text{'}\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash sqrt\; =\; R$.

The Square Root of Two to 5 million digits

by Jerry Bonnell and Robert J. Nemiroff. May, 1994.

Square root of 2 is irrational

a collection of proofs *

Search Engine

2 billion searchable digits of , and {{DEFAULTSORT:Square Root Of Two Quadratic irrational numbers Mathematical constants Pythagorean theorem Articles containing proofs

real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

that, when multiplied by itself, equals the . It may be written in mathematics as $\backslash sqrt$ or $2^$, and is an algebraic number
An algebraic number is any complex number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

. Technically, it should be called the principal square root
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

of 2, to distinguish it from the negative number with the same property.
Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

. It was probably the first number known to be irrational
Irrationality is cognition
Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...

. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

.
Sequence in the On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transferred the intellectual property
Intellectual property (I ...

consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places:
:
History

TheBabylonia
Babylonia () was an and based in central-southern which was part of Ancient Persia (present-day and ). A small -ruled state emerged in 1894 BCE, which contained the minor administrative town of . It was merely a small provincial town dur ...

n clay tablet YBC 7289
YBC 7289 is a Babylonia
Babylonia () was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia
Mesopotamia ( ar, بِلَاد ٱلرَّافِدَيْن '; grc, Μεσοποταμία; Syriac l ...

(c. 1800–1600 BC) gives an approximation of in four sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ...

figures, , which is accurate to about six decimal
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...

digits, and is the closest possible three-place sexagesimal representation of :
:$1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; \backslash frac\; =\; 1.41421\backslash overline.$
Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC), as follows: ''Increase the length f the sideby its third and this third by its own fourth less the thirty-fourth part of that fourth.'' That is,
:$1\; +\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; =\; \backslash frac\; =\; 1.41421\backslash overline.$
This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell number
Pell is a surname shared by several notable people, listed below
* Axel Rudi Pell (born 1960), German heavy metal guitar player and member of Steeler and founder of his own eponymous band
* Charles Pell (1874–1936), American college football c ...

s, which can be derived from the continued fraction
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
Pythagoreans
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...

discovered that the diagonal of a square
In Euclidean geometry, a square is a regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* Regular (Badfinger ...

is incommensurable with its side, or in modern language, that the square root of two is irrational
Irrationality is cognition
Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...

. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus
Hippasus of Metapontum
Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...

of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by .
Ancient Roman architecture

Inancient Roman architecture
, Spain, a masterpiece of ancient bridge building
, a Roman theatre in Athens, Greece
Ancient Roman architecture adopted the external language of classical Ancient Greek Architecture, Greek architecture for the purposes of the ancient Ro ...

, Vitruvius
Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

describes the use of the square root of 2 progression or ''ad quadratum'' technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato
Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thoug ...

. The system was employed to build pavements by creating a square tangent
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

to the corners of the original square at 45 degrees of it. The proportion was also used to design atriaAtria may refer to:
*Atrium (heart)
The atrium (Latin ātrium, “entry hall”) is the upper chamber through which blood enters the Ventricle (heart), ventricles of the heart. There are two atria in the human heart – the left atrium receives bloo ...

by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.
Decimal value

Computation algorithms

There are a number ofalgorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s for approximating as a ratio of integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the Babylonian method
Methods of computing square roots are numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... ...

for computing square roots, which is one of many methods of computing square roots
Methods of computing square roots are numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... ...

. It goes as follows:
First, pick a guess, ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics
Linguistics is the science, scientific study of language. It e ...

computation:
:$a\_\; =\; \backslash frac=\backslash frac+\backslash frac.$
The more iterations through the algorithm (that is, the more computations performed and the greater ""), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with , the results of the algorithm are as follows:
* 1 ()
* = 1.5 ()
* = 1.416... ()
* = 1.414215... ()
* = 1.4142135623746... ()
Rational approximations

A simple rational approximation (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than (approx. ). The next two better rational approximations are (≈ 1.4141414...) with a marginally smaller error (approx. ), and (≈ 1.4142012) with an error of approx . The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with () is too large by about ; its square is ≈ .Records in computation

In 1997 the value of was calculated to 137,438,953,444 decimal places byYasumasa Kanada
was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of pi, . He set the record 11 of the past 21 times.
Kanada was a professor in the Department of Information Science at ...

's team. In February 2006 the record for the calculation of was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion
A trillion is a number with two distinct definitions:
* 1,000,000,000,000, i.e. one million million, or (ten to the twelfth power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted ...

decimal places in 2010. Among mathematical constant
A mathematical constant is a key whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an ), or by mathematicians' names to facilitate using it across multiple s. Constants arise in many areas of , with constan ...

s with computationally challenging decimal expansions, only has been calculated more precisely. Such computations aim to check empirically whether such numbers are normal.
This is a table of recent records in calculating the digits of .
Proofs of irrationality

A shortproof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

of the irrationality of can be obtained from the rational root theorem
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

, that is, if is a monic polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with integer coefficient
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, then any rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογι ...

root
In vascular plant
Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large group ...

of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any square root of any natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

that is not a perfect square is irrational.
For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational numberIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

or Infinite descentIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
Proof by infinite descent

One proof of the number's irrationality is the followingproof by infinite descentIn mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for ...

. It is also a proof by contradiction
In logic and mathematics, proof by contradiction is a form of Mathematical proof, proof that establishes the Truth#Formal theories, truth or the Validity (logic), validity of a proposition, by showing that assuming the proposition to be false leads ...

, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true.
# Assume that is a rational number, meaning that there exists a pair of integers whose ratio is exactly .
# If the two integers have a factor
FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry".
FACTOR was founded in 1982 by r ...

, it can be eliminated using the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

.
# Then can be written as an irreducible fraction
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquial ...

such that and are coprime integers
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

(having no common factor) which additionally means that at least one of or must be odd.
# It follows that and . ( ) ( are integers)
# Therefore, is even because it is equal to . ( is necessarily even because it is 2 times another whole number.)
# It follows that must be even (as squares of odd integers are never even).
# Because is even, there exists an integer that fulfills .
# Substituting from step 7 for in the second equation of step 4: is equivalent to , which is equivalent to .
# Because is divisible by two and therefore even, and because , it follows that is also even which means that is even.
# By steps 5 and 8 and are both even, which contradicts that is irreducible as stated in step 3.
::'' Q.E.D.''
Because there is a contradiction, the assumption (1) that is a rational number must be false. This means that is not a rational number. That is, is irrational.
This proof was hinted at by Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental quest ...

, in his '' Analytica Priora'', §I.23. It appeared first as a full proof in Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's '' Elements'', as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an interpolation
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

and not attributable to Euclid.
Proof by unique factorization

As with the proof by infinite descent, we obtain $a^2\; =\; 2b^2$. Being the same quantity, each side has the same prime factorization by thefundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.
Geometric proof

A simple proof is attributed byJohn Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when con ...

to Stanley Tennenbaum
Stanley Tennenbaum (April 11, 1927 – May 4, 2005) was an American mathematician who contributed to the field of logic. In 1959, he published Tennenbaum's theorem, which states that no countable set, countable Non-standard model of arithmetic, non ...

when the latter was a student in the early 1950s and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of ''American Scientist
__NOTOC__
''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society. In the beginning of 2000s the headquarters was in New H ...

''. Given two squares with integer sides respectively ''a'' and ''b'', one of which has twice the area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.
Another geometric reductio ad absurdum
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...

argument showing that is irrational appeared in 2000 in the American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located i ...

. It is also an example of proof by infinite descent. It makes use of classic compass and straightedge
Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandr ...

construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the same algebraic proof as in the previous paragraph, viewed geometrically in another way.
Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

, . Suppose and are integers. Let be a ratio
In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

given in its lowest terms
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English ...

.
Draw the arcs and with centre . Join . It follows that , and the and coincide. Therefore, the triangle
A triangle is a polygon
In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

s and are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...

by SAS.
Because is a right angle and is half a right angle, is also a right isosceles triangle. Hence implies . By symmetry, , and is also a right isosceles triangle. It also follows that .
Hence, there is an even smaller right isosceles triangle, with hypotenuse length and legs . These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms. Therefore, and cannot be both integers, hence is irrational.
Constructive proof

In a constructive approach, one distinguishes between on the one hand not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given positive integers and , because the (i.e., highest power of 2 dividing a number) of is odd, while the valuation of is even, they must be distinct integers; thus . Then :$\backslash left,\; \backslash sqrt2\; -\; \backslash frac\backslash \; =\; \backslash frac\; \backslash ge\; \backslash frac\; \backslash ge\; \backslash frac,$ the latterinequality
Inequality may refer to:
Economics
* Attention inequality
Attention inequality is a term used to target the inequality of distribution of attention across users on social networks, people in general, and for scientific papers. Yun Family Foundat ...

being true because it is assumed that (otherwise the quantitative apartness can be trivially established). This gives a lower bound of for the difference , yielding a direct proof of irrationality not relying on the law of excluded middle
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...

; see Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of America ...

(1985, p. 18). This proof constructively exhibits a discrepancy between and any rational.
Proof by Diophantine equations

* ''Lemma'': For theDiophantine equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

$x^2+y^2=z^2$ in its primitive (simplest) form, integer solutions exist if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

either $x$ or $y$ is odd, but never when both $x$ and $y$ are odd.
''Proof'': For the given equation, there are only six possible combinations of oddness and evenness for whole-number values of $x$ and $y$ that produce a whole-number value for $z$. A simple enumeration of all six possibilities shows why four of these six are impossible. Of the two remaining possibilities, one can be proven to not contain any solutions using modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...

, leaving the sole remaining possibility as the only one to contain solutions, if any.
The fifth possibility (both $x$ and $y$ odd and $z$ even) can be shown to contain no solutions as follows.
Since $z$ is even, $z^2$ must be divisible by $4$, hence
:$x^2+y^2\; \backslash equiv\; 0\; \backslash mod4$
The square of any odd number is always congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...

to 1 modulo 4. The square of any even number is always congruent to 0 modulo 4. Since both $x$ and $y$ are odd and $z$ is even:
:$1+1\; \backslash equiv\; 0\; \backslash mod\; 4$
:$2\; \backslash equiv\; 0\; \backslash mod\; 4$
which is impossible. Therefore, the fifth possibility is also ruled out, leaving the sixth to be the only possible combination to contain solutions, if any.
An extension of this lemma is the result that two identical whole-number squares can never be added to produce another whole-number square, even when the equation is not in its simplest form.
* ''Theorem:'' $\backslash sqrt2$ is irrational.
''Proof'': Suppose to the contrary $\backslash sqrt2$ is rational. Therefore,
:$\backslash sqrt2\; =$
:where $a,b\; \backslash in\; \backslash mathbb$ and $b\; \backslash neq\; 0$
:Squaring both sides,
:$2\; =$
:$2b^2\; =\; a^2$
:$b^2+b^2\; =\; a^2$
But the lemma proves that the sum of two identical whole-number squares cannot produce another whole-number square.
Therefore, the assumption that $\backslash sqrt2$ is rational is contradicted.
$\backslash sqrt2$ is irrational. '' Q. E. D.''
Multiplicative inverse

Themultiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

(reciprocal) of the square root of two (i.e., the square root of ) is a widely used constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...

.
:$\backslash frac1\; =\; \backslash frac\; =\; \backslash sin\; 45^\backslash circ\; =\; \backslash cos\; 45^\backslash circ\; =$ ...
One-half of , also the reciprocal of , is a common quantity in geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and trigonometry
Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

because the unit vector
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

that makes a 45° angle
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

with the axes in a plane has the coordinates
:$\backslash left(\backslash frac,\; \backslash frac\backslash right)\backslash !.$
This number satisfies
:$\backslash tfrac\backslash sqrt\; =\; \backslash sqrt\; =\; \backslash frac\; =\; \backslash cos\; 45^\; =\; \backslash sin\; 45^.$
Properties

One interesting property of is :$\backslash !\backslash \; =\; \backslash sqrt\; +\; 1$ since :$\backslash left(\backslash sqrt+1\backslash right)\backslash !\backslash left(\backslash sqrt-1\backslash right)\; =\; 2-1\; =\; 1.$ This is related to the property ofsilver ratio
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

s.
can also be expressed in terms of copies of the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...

using only the square root
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

and arithmetic operations
Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...

, if the square root symbol is interpreted suitably for the complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s and :
:$\backslash frac\backslash text\backslash frac$
is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for , and for , the limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

of as will be called (if this limit exists) . Then is the only number for which . Or symbolically:
:$\backslash sqrt^\; =\; 2.$
appears in for :
: $2^m\backslash sqrt\; \backslash to\; \backslash pi\backslash textm\; \backslash to\; \backslash infty$
for square roots and only one minus sign.
Similar in appearance but with a finite number of terms, appears in various trigonometric constants:
:$\backslash begin\; \backslash sin\backslash frac\; \&=\; \backslash tfrac12\backslash sqrt\; \&\backslash quad\; \backslash sin\backslash frac\; \&=\; \backslash tfrac12\backslash sqrt\; \&\backslash quad\; \backslash sin\backslash frac\; \&=\; \backslash tfrac12\backslash sqrt\; \backslash \backslash $\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt &\quad
\sin\frac &= \tfrac12\sqrt
\end
It is not known whether is a normal numberNormal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Normal ...

, which is a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.
Representations

Series and product

The identity , along with the infinite product representations for thesine and cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

, leads to products such as
:$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash left(1-\backslash frac\backslash right)\; =\; \backslash left(1-\backslash frac\backslash right)\backslash !\backslash left(1-\backslash frac\backslash right)\backslash !\backslash left(1-\backslash frac\backslash right)\; \backslash cdots$
and
:$\backslash sqrt\; =\; \backslash prod\_^\backslash infty\backslash frac\; =\; \backslash left(\backslash frac\backslash right)\backslash !\backslash left(\backslash frac\backslash right)\backslash !\backslash left(\backslash frac\backslash right)\backslash !\backslash left(\backslash frac\backslash right)\; \backslash cdots$
or equivalently,
:$\backslash sqrt\; =\; \backslash prod\_^\backslash infty\backslash left(1+\backslash frac\backslash right)\backslash left(1-\backslash frac\backslash right)\; =\; \backslash left(1+\backslash frac\backslash right)\backslash !\backslash left(1-\backslash frac\backslash right)\backslash !\backslash left(1+\backslash frac\backslash right)\backslash !\backslash left(1-\backslash frac\backslash right)\; \backslash cdots.$
The number can also be expressed by taking the Taylor series
In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...

of a trigonometric function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. For example, the series for gives
:$\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$
The Taylor series of with and using the double factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

gives
:$\backslash sqrt\; =\; \backslash sum\_^\backslash infty\; (-1)^\; \backslash frac\; =\; 1\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; =\; 1\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$
The convergence
Convergence may refer to:
Arts and media Literature
*Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-par ...

of this series can be accelerated with an Euler transformIn combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the ...

, producing
:$\backslash sqrt\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$
It is not known whether can be represented with a BBP-type formula. BBP-type formulas are known for and , however.
The number can be represented by an infinite series of Egyptian fractions
An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, ...

, with denominators defined by 2Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathem ...

-like recurrence relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

''a''(''n'') = 34''a''(''n''−1) − ''a''(''n''−2), ''a''(0) = 0, ''a''(1) = 6.
:$\backslash sqrt=\backslash frac-\backslash frac\backslash sum\_^\backslash infty\; \backslash frac=\backslash frac-\backslash frac\backslash left(\backslash frac+\backslash frac+\backslash frac+\backslash dots\; \backslash right)$
Continued fraction

The square root of two has the followingcontinued fraction
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

representation:
:$\backslash !\backslash \; \backslash sqrt\; =\; 1\; +\; \backslash cfrac.$
The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell number
Pell is a surname shared by several notable people, listed below
* Axel Rudi Pell (born 1960), German heavy metal guitar player and member of Steeler and founder of his own eponymous band
* Charles Pell (1874–1936), American college football c ...

s (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: . The convergent differs from by almost exactly and then the next convergent is .
Nested square

The following nested square expressions converge to : :$\backslash begin\; \backslash sqrt\; \&=\backslash tfrac\; -\; 2\; \backslash left(\; \backslash tfrac-\; \backslash left(\; \backslash tfrac-\backslash left(\; \backslash tfrac-\; \backslash left(\; \backslash tfrac-\; \backslash cdots\; \backslash right)^2\; \backslash right)^2\; \backslash right)^2\; \backslash right)^2\backslash \backslash \; \&=\backslash tfrac\; -\; 4\; \backslash left(\; \backslash tfrac+\; \backslash left(\; \backslash tfrac+\backslash left(\; \backslash tfrac+\; \backslash left(\; \backslash tfrac+\; \backslash cdots\; \backslash right)^2\; \backslash right)^2\; \backslash right)^2\; \backslash right)^2.\; \backslash end$Applications

Paper size

In 1786, German physics professorGeorg Christoph Lichtenberg
Georg Christoph Lichtenberg (1 July 1742 – 24 February 1799) was a German physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area ...

found that any sheet of paper whose long edge is times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised paper size
Paper size Technical standard, standards govern the size of sheets of paper used as writing paper, stationery, cards, and for some printed documents.
The ISO 216 standard, which includes the commonly used A4 size, is the international standa ...

s at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes. Today, the (approximate) aspect ratio
The aspect ratio of a geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμό ...

of paper sizes under ISO 216
ISO 216 is an International Organization for Standardization, international standard for paper sizes, used around the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, incl ...

(A4, A0, etc.) is 1:.
Proof:Let $S\; =$ shorter length and $L\; =$ longer length of the sides of a sheet of paper, with

:$R\; =\; \backslash frac\; =\; \backslash sqrt$ as required by ISO 216. Let $R\text{'}\; =\; \backslash frac$ be the analogous ratio of the halved sheet, then

:$R\text{'}\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash sqrt\; =\; R$.

Physical sciences

There are some interesting properties involving the square root of 2 in thephysical sciences
Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences".
...

:
* The square root of two is the frequency ratio
9:8 major tone
In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent sta ...

of a tritone
In music theory
Music theory is the study of the practices and possibilities of music
Music is the art of arranging sounds in time through the elements of melody, harmony, rhythm, and timbre. It is one of the universal cultural aspects ...

interval in twelve-tone equal temperament
An equal temperament is a or , which approximates by dividing an (or other interval) into equal steps. This means the ratio of the of any adjacent pair of notes is the same, which gives an equal perceived step size as is perceived roughly a ...

music.
* The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of ''areas'' between two successive aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of ray (optics), rays that come to a focus (optics), focus ...

s is 2.
* The celestial latitude (declination) of the Sun during a planet's astronomical cross-quarter day points equals the tilt of the planet's axis divided by .
See also

*List of mathematical constantsA mathematical constant
A mathematical constant is a key number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. ...

*Square root of 3
The square root of 3 is the positive real number
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican ...

,
* Square root of 5
The square root of 5 is the positive real number
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican ...

,
* Gelfond–Schneider constant,
* Silver ratio
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

,
Notes

References

* . * * Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI. * . * . * . * .External links

* .The Square Root of Two to 5 million digits

by Jerry Bonnell and Robert J. Nemiroff. May, 1994.

Square root of 2 is irrational

a collection of proofs *

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